AbInitio …authors.library.caltech.edu/60476/1/PhysRevLett.112... · 2015-09-24 · functional theory and many-body perturbation theory to investigate hot carriers in semiconductors.

Ab Initio Study of Hot Carriers in the First Picosecond after Sunlight Absorption in Silicon

Marco Bernardi,1,2 Derek Vigil-Fowler,1,2 Johannes Lischner,1,2 Jeffrey B. Neaton,1,2,3,4,* and Steven G. Louie1,2,†1Department of Physics, University of California, Berkeley, California 94720-7300, USA

2Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA3Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

4Kavli Institute for Energy Nanosciences, Berkeley, California 94720, USA(Received 21 January 2014; published 26 June 2014)

Hot carrier thermalization is a major source of efficiency loss in solar cells. Because of thesubpicosecond time scale and complex physics involved, a microscopic characterization of hot carriersis challenging even for the simplest materials. We develop and apply an ab initio approach based on densityfunctional theory and many-body perturbation theory to investigate hot carriers in semiconductors. Ourcalculations include electron-electron and electron-phonon interactions, and require no experimental inputother than the structure of the material. We apply our approach to study the relaxation time and mean freepath of hot carriers in Si, and map the band and k dependence of these quantities. We demonstrate that a hotcarrier distribution characteristic of Si under solar illumination thermalizes within 350 fs, in excellentagreement with pump-probe experiments. Our work sheds light on the subpicosecond time scale aftersunlight absorption in Si, and constitutes a first step towards ab initio quantification of hot carrier dynamicsin materials.

DOI: 10.1103/PhysRevLett.112.257402 PACS numbers: 78.56.-a, 71.20.Mq, 78.47.db, 88.40.H-

Single-junction solar cells based on crystalline Si arerapidly approaching the Shockley-Queisser efficiency limit[1,2]. While the Carnot efficiency of ∼95% sets theultimate limit for solar energy conversion at room temper-ature, practical efficiency limits in ordinary photovoltaic(PV) solar cells are significantly lower; e.g., the Shockley-Queisser limit for Si is close to 30% [2]. The main factorslimiting efficiency are carrier thermalization and absorptionlosses [3,4]. For the case of Si under AM1.5 solarillumination [5], nearly 25% of incident solar energy islost to heat as the nonequilibrium (“hot”) carriers generatedby sunlight absorption thermalize to the edges of the bandgap. Not only is hot carrier thermalization the main sourceof loss in most PV materials, it is also difficult to prevent,control, and understand with microscopic detail due to thesubpicosecond time scale typical of hot carrier relaxation[6]. This scenario is common to other technologies employ-ing hot carriers, including electronics, optoelectronics, andrenewable energy devices beyond PV [7–11].The leading mechanisms involved in hot carrier thermali-

zation consist of inelastic electron-phonon (e-ph) andelectron-electron (e-e) scattering processes [12]. Relaxationtimes for e-ph and e-e interactions in semiconductorshave been studied extensively by model Hamiltonians withselected phonon modes, simplified electronic band struc-tures, deformation potentials, and/or empirical pseudopo-tentials [13–18]. Hot carrier dynamics in semiconductorshas been investigated experimentally using pump-probeoptical measurements [19,20].This work has two main goals. First, we present an

ab initio approach based on density functional theory

(DFT) and many-body perturbation theory to investigatehot carriers in materials. Second, we apply this framework tostudy hot carrier thermalization at subpicosecond time scaleafter sunlight absorption in the simplest andmost commonlyused material in PV, namely Si. Our approach improves overprevious methods employed to study hot carriers [16–18],and is widely applicable. In particular, our formalism allowsone to (1) treat all phonon modes on the same footing byusing full phonon dispersion curves rather than singlephononmodes, (2) resolve hot carrier relaxation times, meanfree paths, and dynamics for different bands and k points inthe Brillouin zone (BZ), and (3) compute the e-ph couplingmatrix elements on fine grids in the BZ by combiningab initio pseudopotentials and wave functions. For example,within our ab initio framework there is no need to employdeformation potentials to describe e-ph scattering rates.This work does not aim to suggest strategies to mitigatethermalization losses in solar cells [21,22], but rather aimsto study hot carriers using accurate ab initio calculations.We compute hot carrier relaxation times and mean free

paths in Si, and map these quantities in the BZ. Ourcalculations indicate that hot carrier thermalization in Si isdominated by phonon emission processes with a time scaleof ∼100 fs for carriers near the band edges, while fasterrelaxation times of ∼10 fs are found at higher energiesaway from the band edges. We predict that an initial carrierdistribution characteristic of Si under AM1.5 solar illumi-nation thermalizes within 350 fs, in excellent agreementwith pump-probe experiments.We carry out calculations on a unit cell of Si with a

lattice parameter of 5.4 Å. The ground-state electronic

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structure is computed within the local density approxima-tion (LDA) of DFT using the QUANTUM ESPRESSO code[23,24]. Norm-conserving pseudopotentials are employed[25], and a kinetic energy cutoff of 45 Ry is used forthe plane-wave basis set. Lattice-dynamical properties are

computed using density functional perturbation theory[26]. We employ the EPW code [27] to compute theimaginary part of the e-ph self-energy ImðΣe-ph

n;k Þ forthe Bloch state of energy ϵn;k at band n and k point inthe BZ [28]:

ImðΣe-phnk Þ ¼

Xm;λ;q

jgλ;qn;m;kj2Im�

Nλ;q þ 1 − fm;k

ϵn;k − ϵm;kþq − ℏωλ;q − iηþ Nλ;q − fm;k

ϵn;k − ϵm;kþq þ ℏωλ;q − iη

�; ð1Þ

where ℏωλ;q is the energy of a phonon with polarization λand wave vector q in the BZ, fm;k and Nλ;q are Fermi andBose occupation numbers, respectively, evaluated here at300 K, and η is a small Lorentzian broadening. In Eq. (1),the key quantities are the e-ph matrix elements, defined as

gλ;qn;m;k ¼ hΨm;kþqj∂λ;qVjΨn;ki; ð2Þ

where Ψn;k is a Kohn-Sham wave function, and ∂λ;qV isthe variation of the Kohn-Sham potential for a unitdisplacement of the nuclei along the phonon mode ofpolarization λ and wave vector q. The e-ph relaxation timesare computed from the imaginary part of the self-energy asτe-phn;k ¼ ½ImðΣe-ph

n;k Þ�−1ℏ=2. We first compute the electronicand vibrational states and the associated e-ph matrixelements on 12 × 12 × 12 k and 6 × 6 × 6 q point gridsusing DFT. We then interpolate the quantities needed toevaluate the e-ph self-energy on significantly finer grids of50 × 50 × 50 k and q points, using an interpolation pro-cedure based on Wannier functions [29,30] implemented inthe EPW code [27]. Our Wannier functions consist of foursp3 orbitals on each Si atom, leading to the wannierizationof 4 valence and 4 conduction bands. We carry out full-frequency GW calculations [31] using the BERKELEY-GWcode [32] to compute the imaginary part of the e-e self-energy, ImðΣe-e

n;kÞ, and the associated e-e relaxation time

τe-en;k ¼ ½ImðΣe-phn;k Þ�−1ℏ=2. Kinetic energy cutoffs of 5 and

10 Ry are used, respectively, for the screened and bareCoulomb interactions, and 96 empty bands are used tocompute the dielectric screening and the Green’s function[33]. Finally, we study the time evolution of the electronicoccupation number fn;kðtÞ for hot carriers. At each time t,the occupations are time stepped using the relaxation-timeapproximation of the Boltzmann equation:

dfn;kðtÞdt

¼ −fn;kðtÞ − fn;kðtthÞ

τn;k; ð3Þ

where tth is the time after which thermalization is complete,and the relaxation time τn;k is obtained by combining thee-ph and e-e relaxation times via the Matthiessen rule,ðτn;kÞ−1 ¼ ðτe-en;kÞ−1 þ ðτe-phn;k Þ−1.

Figure 1(a) shows the imaginary part of the e-ph and e-eself-energies in the energy range of relevance for solarexcitation [5]. We find that the e-ph contribution to theimaginary part of the self-energy is significantly larger thanthe e-e contribution throughout the energy range of interest.In addition, ImðΣe-phÞ shows a strong energy dependencewith trends similar to the electronic density of states (DOS),consistent with the fact that in this case the DOS regulatesthe phase space for e-ph scattering [14]. The hot electronscattering rates in Fig. 1(a) are in very good agreement withprevious calculations from Fischetti et al. using empiricalpseudopotentials [16,17]. Our ability to resolve e-ph proc-esses on fine grids in the BZ leads to additional details notobserved in previous calculations. For example, we observethe presence of a feature with lower ImðΣe-phÞ values(equivalently, lower scattering rates) due to the L valley at∼1 eV above the conduction band edge, as well as aninherent spread of the scattering rate for a fixed energy due tothe k dependence of the e-ph matrix elements. The e-e self-energy is smaller than 10meVwithin 3 eVof themidpoint ofthe band gap, and vanishes for energies between Eg belowthe valence band maximum (VBM) and Eg above theconduction band minimum (CBM), where Eg ¼ 1.15 eVis the band gap of Si. These trends can be understood fromthe fact that ourGW self-energy encodes information relatedto the electron-hole processes of a quasiparticle at zerotemperature. Hence, the lowest-energy electron-hole

FIG. 1 (color online). (a) Imaginary part of the e-ph and e-eself-energies, and the electronic DOS. (b) Relaxation timesassociated with the e-ph self-energy alone and with both thee-ph and e-e self-energies (curve labeled as “total”). The zero ofthe energy axis is placed at the midpoint of the band gap (shownas a shaded area).

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channels captured here consist of interband impact ioniza-tion and Auger recombination processes, the onset of whichis∼Eg below theVBMand above theCBM, respectively, forhole and electron quasiparticles. Because of the low carrierconcentrations excited by solar irradiation—typically lessthan 1017 cm−3 carriers—intraband e-e processes in thehot carrier gas are expected to play a minor role [19]. Weconclude that for Si under solar excitation, hot carrierthermalization is dominated by phonon emission processes,while impact ionization processes are less effective due tothe 1.15 eV gap of Si and the low carrier concentration.Our data in Fig. 1(b) further indicate that hot carrier

relaxation near the band edges is characterized by relax-ation times of 50–100 fs, while faster relaxation times of10–20 fs are found at energies more than 300 meV awayfrom the band edges, for both electrons and holes. Theinclusion of e-e interactions changes the relaxation timesonly very slightly [Fig. 1(b)]. Within an optical phononenergy (∼60 meV in Si) of the VBM and CBM, therelaxation times become larger than 100 fs due to the lackof phonon emission processes. In this energy rangethermalization is typically fast and mediated by intrabande-e processes not included here. The rate of hot carrierthermalization is thus limited by the slowest e-ph relaxationtimes of 50–100 fs at energies of ∼200 meV away from theband edges.The rate of e-ph processes responsible for hot carrier

thermalization can be mapped in the BZ by analyzingImðΣe-ph

n;k Þ along high-symmetry directions. Figure 2 showsthe electronic band structure of Si combined with infor-mation related to ImðΣe-ph

n;k Þ on a fine k grid. We observe a

strong k and band dependence of ImðΣe-phn;k Þ and the

associated e-ph relaxation time τe-phn;k . Energy windowswith several bands result in shorter relaxation times dueto a larger number of possible transitions accompanyingphonon emission from hot carriers. The k dependence ofthe relaxation time is further dictated by the k dependenceof the e-ph matrix elements.Maps of e-ph scattering rates in the BZmay prove crucial

to devise new strategies for hot carrier utilization [11,21].

By relating high-symmetry directions in the BZ to realspace, information regarding hot carrier relaxation alongdifferent crystallographic directions can be retrieved, thusassisting the design of experiments aimed at collecting hotcarriers. We combine the semiclassical velocities vn;k ¼ℏ−1∂En;k=∂k with the e-ph relaxation times to obtain thee-ph mean free paths (MFPs) Ln;k ¼ vn;k · τn;k. The e-phMFPs represent the average distance hot carriers travelbefore emitting (or absorbing) a phonon. When collected atdistances shorter than the MFP, the energy and momentumdistributions of hot carriers are preserved, thus enablingextraction before thermalization [7,11].Figure 3 shows e-ph MFPs for hot electrons and holes in

Si along the [100], [110], and [111] directions, obtained bycomputing vn;k and τn;k on fine grids, respectively, alongthe ΓX, ΓK, and ΓL directions of the BZ. For hot electrons,we observe MFPs of less than 5 nm at all energies and forall directions, with the exception of the [100] direction atenergies near the CBM for which we predict MFPs of up to15 nm. Such longer MFPs are obtained for energies upto 0.5 eV above the CBM, and stem from velocities nearthe bottom of the X valley, combined with relatively long(50–100 fs) relaxation times due to small DOS values nearthe CBM. Hot holes exhibit overall larger MFPs of up to15 nm that are evenly distributed among the differentdirections. Our results indicate that extraction of hotelectrons is best achieved in a thin film of Si(100) withless than ∼10 nm thickness, combined with excitationenergies of ∼1.5 eV. On the other hand, we predict thathot holes may be extracted from thin films of up to 15 nmthickness with any orientation for excitation energies in the1.2–3 eV range [34].We use the data obtained so far to simulate the ultrafast

thermalization of 1017 cm−3 hot carriers generated bysunlight absorption in Si [35]. First, the initial hot carrieroccupations fn;kðt ¼ 0Þ imparted by sunlight are estimatedby combining the computed density of states DðEÞ withexperimental data for optical absorption in Si [36] andAM1.5 solar radiation [5]:

FIG. 2 (color online). The band structure of Si, together with acolor map of the ImðΣe-phÞ and relaxation time.

FIG. 3 (color online). Mean free path as a function of energyfor hot holes (a) and electrons (b) in Si, shown for the threecrystallographic directions [100], [110], and [111]. The curvescombine data from multiple bands. The zero of the energy axis isplaced at the midpoint of the band gap.

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fn;kðt¼ 0Þ∝Z

4 eV

0

dωDðϵn;k�ωÞJphðωÞαðωÞθðω� ϵn;kÞ;

ð4Þ

where ϵn;k is the energy of the hot carrier, ω is the photonenergy, Jph is the solar irradiance, α is the absorptionspectrum of Si, θ is the theta step function, and the uppersign refers to holes while the lower sign refers to electrons.In Eq. (4), the band index n runs over the valence andconduction bands for hot holes and electrons, respectively,and the equation estimates the number of interband transi-tions generating hot carriers. The occupations are thentime stepped using Eq. (3), with the occupations fn;kðtthÞafter thermalization consisting of a Fermi-Dirac distribu-tion centered at an electron quasi-Fermi level such that1017 cm−3 carriers are still present after thermalization.This approach is consistent with the fact that the time scalefor thermalization is much shorter than the ∼100 μsinterband recombination time in Si [37].Figures 4(a) and 4(b) show the hot carrier population,

PðEÞ ¼ fðEÞDðEÞ, for holes and electrons at times t ¼ 0and after thermalization. Since the states within 300 meVof the band edges possess relaxation time one order ofmagnitude longer than states at higher energy within thebands, we expect the thermalization rate to be limited bythe slower dynamics of hot carriers near the band edges.

The thermalization of hot carriers in this energy range isdetailed in Figs. 4(c) and 4(d), showing the carrierpopulations as a function of time together with the finalFermi-Dirac distribution for thermalized carriers. Theenergy rangewithin∼60 meV of the band edges is excludedas thermalization there is not contributed by e-ph processes,as noted above. Our results show that thermalization nearthe band edges completes in ∼200 fs for holes and ∼250 fsfor electrons. Thermalization completes within 50 fs athigher energies within the bands (not shown), consistentwith the faster relaxation times of ∼10 fs predicted forenergies away from the band edges.We compare our results with experiments for hot electron

thermalization in Si. Doany et al. [20] performed time-resolved reflectivity measurements on Si, and obtained athermalization time of 350 fs for 1017 cm−3 hot electronsexcited with 0.8 eV excess energy above the CBM. Ourresults are in excellent agreement with their measurements.In fact, we predict thermalization times of 250 fs at energiesnear the conduction band edge, combined with a loss of∼60 meV (the energy of an optical phonon in Si) every∼10 fs at higher energies. For hot carriers with an energy of0.8 eVabove the CBM as in Ref. [20], we thus predict a lossof 600 meV in 100 fs to relax the hot carriers to 200 meVabove the CBM, followed by thermalization in 250 fs.This yields a total thermalization time of approximately350 fs, as in the experiments in Ref. [20]. Our results arealso consistent with thermalization times extracted fromtransport measurements in combination with Monte Carlocalculations [18,20,38], predicting thermalization within250 fs near the CBM. Since the vastmajority of hot electronsgenerated by sunlight in Si are found within 1–2 eV abovethe CBM, we conclude that hot electron thermalizationunder sunlight illumination in Si occurs over ∼350 fs. Ourcalculations indicate a similar time scale for hot holethermalization.In summary, our work demonstrates that hot carrier

thermalization losses in Si under sunlight illuminationoccur at ultrafast (∼350 fs) time scale and are dominatedby phonon emission from hot carriers with energies nearthe band edges. Our ab initio approach can providemicroscopic information regarding hot carriers that wouldbe difficult to extract from experiment, such as the energyand momentum dependence of hot carrier relaxation timesand MFPs. Our work sheds light on the microscopic originof the main efficiency loss in Si solar cells, and paves theway to ab initio studies of hot carriers in materials forrenewable energy.

M. B. thanks Sinisa Coh for discussion. This researchwas supported by the SciDAC Program on Excited StatePhenomena in Energy Materials funded by the U.S.Department of Energy, Office of Basic Energy Sciencesand of Advanced Scientific Computing Research, underContract No. DE-AC02-05CH11231 at Lawrence BerkeleyNational Laboratory, which provided for algorithm and

FIG. 4 (color online). Initial hot carrier population (red curve)after photoexcitation with AM1.5 sunlight, and final populationafter thermalization (blue curve) for 1017 cm−3 holes (a) andelectrons (b) in Si. The zero of the energy axis is placed at themidpoint of the band gap. The hot hole (c) and electron (d)population dynamics is shown for energies within 300 meVof theVBM and CBM, respectively. Energies within 60 meV ofthe band edges are not shown. The dashed line labeled as “th”is the occupation after thermalization is complete.

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code developments and simulations; and by the NationalScience Foundation under Grant No. DMR 10-1006184which provided for basic theory and formalism. Work at theMolecular Foundry was supported by the Office of Science,Office of Basic Energy Sciences, of the U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231.S. G. L. acknowledges support of a Simons FoundationFellowship in Theoretical Physics. This research usedresources of the National Energy Research ScientificComputing Center, which is supported by the Office ofScience of the U.S. Department of Energy.

*Corresponding [email protected]

†Corresponding [email protected]

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